Question 12205: Can you please help to answer the following question:
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Sum the series:
1 x 2 x 3 - 2^3 + 3 x 4 x 5 - 4^3 + ... + 9999 x 10000 x 10001 x - 10000^3
Answer by khwang(438)   (Show Source):
You can put this solution on YOUR website! To find the sum S =
1 x 2 x 3 - 2^3 + 3 x 4 x 5 - 4^3 + ... + 9999 x 10000 x 10001 x - 10000^3
Consider the general form of every two consecutive terms as:
1 * 2 * 3 - 2^3 = 2(1*3 - 2*2) = -2,
3 * 4 * 5 - 4^3 = 4(3*5 - 4^2) = -4,
...
(2n-1)2n(2n+1) - (2n)^3= 2n[(2n-1)(2n+1) - 4n^2) = 2n(4n^2-1- 4n^2) = -2n,
....
9999 * 10000 * 10001 - 10000^3 = - 10000 (last two terms)
Totally, there are 10000/2 = 5000 such pairs.
Hence, S = -(2+4+6+...+10000) = -2(1+2+3+...+5000)
= -2(5001*5000/2) [why ?]
= -25005000
Not really hard.
Kenny
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